This preview shows page 1. Sign up to view the full content.
Unformatted text preview: tile 85 (0.185) median 328 (0.63) 0.95fractile 775 (0.185) CHAPTER 10. USING DATA Using Data to Fit Theoretical
Probability Models Slide No. 21
ENCE 627 ©Assakkaf Method A: One way to deal with data is
simply to fit a theoretical distribution to
it.
Step 1: Decide what kind of distribution is
appropriate (binomial, Poisson, normal,
and so on)
• What distribution is best? Need to understand
the setting
– Defects maybe Poisson
– Value in [0,1] maybe beta
– Normal? Need symmetry as well as other things 11 Slide No. 22 CHAPTER 10. USING DATA Using Data to Fit Theoretical
Probability Models ENCE 627 ©Assakkaf Step 2: Choose the values of the
distribution parameters
• Having chosen the distribution, need to
calibrate, i.e., choose the values for the
parameters. Bernoulli (P), Binomial (n, p),
Poisson ( λ), etc.
• Easy way (probably adequate in a number of
settings.
• Take sample mean and sample variance:
X= Statistical reasons
Statistical reasons
why not n
why not n 1n
∑ xi
n i =1 S2 = 1n
∑ ( xi − X ) 2
n − 1 i =1 Slide No. 23 CHAPTER 10. USING DATA Using Data to Fit Theoretical
Probability Models ENCE 627 ©Assakkaf Example: Calculate the sample mean
(x) and sample variance (S2) for the 35
halfway house observations n = 35
1n
X = ∑ xi = 380.4
n i =1
S2 = ( 1n
∑ xi − X
n − 1 i= ) 2 = 47,344.3 S = 47,344.3 = 217.6 We might choose a normal
distribution
with mean µ = 380.4 and
standard deviation
σ = 217.6 to represent the
distribution of the yearly
bedrental costs. 12 CHAPTER 10. USING DATA Using Data to Fit Theoretical
Probability Models Slide No. 24
ENCE 627 ©Assakkaf Method B: Fit a theoretical distribution
using fractiles. That is, find a theoretical
distribution whose fractiles match as
well as possible with the fractiles of the
empirical data. In this case we would
be fitting a theoretical distribution to a
database distribution. CHAPTER 10. USING DATA Using Data to Fit Theoretical
Probability Models Slide No. 25
ENCE 627 ©Assakkaf Method C: For most initial attempts to
model uncertainty in a decision analysis, it
may be adequate to use the sample mean
and variance as estimates of the mean
and variance of the theoretical distribution
and to establish parameter values in this
way. Refinement of the probability model
may require more careful judgment about
the kind of distribution as well as more
care in fitting the parameters. 13 CHAPTER 10. USING DATA Slide No. 26
ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution When a theoretical distribution has been
assumed, the validity of the assumed
distribution may be verified or disproved
statistically by goodnessoffit tests.
Two tests are commonly used:
– The Chisquare
– The KolmogorovSmirnov test CHAPTER 10. USING DATA Slide No. 27
ENCE 627 ©Assakkaf Testing the Validity of Assumed
Distribution Chisquare Test for Goodness of Fit
– Consider a sample of O observed values of
a random variable.
– The chisquare goodnessoffit test
compares the observed frequencies O1,
O2,…, Ok of k values (k intervals) of the
variate with the corresponding frequencies
E1, E2,…,Ek...
View
Full
Document
This note was uploaded on 10/24/2012 for the course ESI 6385 taught by Professor Mansoorehmollaghasemi during the Fall '12 term at University of Central Florida.
 Fall '12
 MansoorehMollaghasemi

Click to edit the document details